Data-Generating Process

FCMs as a Measure of Stress

FCM Data

Hunting Events

Movement Data

• Contains the location of the 40 collared deer from Feb. 2020 to Feb. 2023
• Movement is tracked at hourly intervals

Limited Data, Large Uncertainty

Limited Data, Large Uncertainty

Other sources of uncertainty include:

• unknown characteristics of the deer (e.g., age, health, etc.),

• other unknown stressors (e.g., predators, human activities, weather, etc.),

• unknown geographical features (e.g., terrain could affect the propagation of sound).

Distance Approximation

Deer location at the time of hunting event is approximated by linear interpolation:

Relevant Hunting Events

A hunting event is considered relevant to an FCM sample, if

• the time difference between defecation and hunting is bounded within gut retention time (GRT) thresholds [hours], and

• the spatial distance between deer and hunting is below a distance threshold [km].

In the following, GRT thresholds = (0, 50), distance threshold = 10 or 15.

The Most Relevant Hunting Event

Among the relevant hunting events, the most relevant one is defined by one of the three introduced proximity criteria:

• the closest in time to GRT target = 19 hours (“closest in time”)
• the closest in space (“nearest”)
• the one with the “highest score”

Illustration

The Scoring Function

we define the Scoring function as following:

\[ S(d, t) \propto \begin{cases} \frac{1}{d^2} \cdot f_\textbf{t}(t), t \sim \mathcal{N}(\mu, \sigma^2) &|t \leq \mu \\ \frac{1}{d^2} \cdot f_\textbf{t}(t), t \sim \mathcal{Laplace}(\mu, b) &|t > \mu \end{cases} \] where:

\[ \begin{align*} d & \text{: Distance } \\ t & \text{: Time Difference } \\ \mu & \text{: GRT target = 19 hours } \end{align*} \]

The Scoring Function

The marginal effects of distance and elapsed time since challenge on the score:

The Models

For Modelling, we consider the following covariates, defined for each pair of FCM sample and most relevant hunting event:

The Models

We chose two different approaches to Modelling:

1. Machine Learning: a model, which focuses on prediction, in our case a XGBoost Model
2. Statistical Modelling: a model, which helps to understand the effects of our covariables, here a General Additive Mixed Model

A. XGBoost

TBD

B. Generalized Additive Mixed Model

• Family: Gamma

• Log link for interpretability

• Let \(i = 1,\dots,N\) be the indices of deer and \(j = 1,\dots,n_i\) be the indices of FCM measurements for each deer

\[ \begin{eqnarray} \textup{FCM}_{ij} &\sim& \mathcal{Ga}\left( \nu, \frac{\nu}{\mu_{ij}} \right) \\ \mu_{ij} &=& \mathbb{E}(\textup{FCM}_{ij}) = \exp(\eta_{ij}) \\ \eta_{ij} &=& \beta_0 + \beta_1 \textup{Pregnant}_{ij} + \beta_2 \textup{NumberOtherHunts}_{ij} + \\ && f_1(\textup{TimeDiff}_{ij}) + f_2(\textup{Distance}_{ij}) + \\ && f_3(\textup{SampleDelay}_{ij}) + f_4(\textup{DefecationDay}_{ij}) + \\ && \gamma_{i}, \\ \gamma_i &\overset{\mathrm{iid}}{\sim}& \mathcal{N}(0, \sigma_\gamma^2). \end{eqnarray} \]

B Generalized Additive Mixed Model

Closest in Time

B Generalized Additive Mixed Model

Nearest

B Generalized Additive Mixed Model

Highest score

B Coefficient Table

Linear Effects: